lattice dynamics calculations based on density functional perturbation theory in real space

Accepted Manuscript Lattice dynamics calculations based on density-functional perturbation theory in real space Honghui Shang, Christian Carbogno, Patrick Rinke, Matthias Scheffler PII: DOI: Reference: S0010-4655(17)30043-7 http://dx.doi.org/10.1016/j.cpc.2017.02.001 COMPHY 6149 To appear in: Computer Physics Communications Received date: 12 October 2016 Revised date: February 2017 Accepted date: February 2017 Please cite this article as: H Shang, C Carbogno, P Rinke, M Scheffler, Lattice dynamics calculations based on density-functional perturbation theory in real space, Computer Physics Communications (2017), http://dx.doi.org/10.1016/j.cpc.2017.02.001 This is a PDF file of an unedited manuscript that has been accepted for publication As a service to our customers we are providing this early version of the manuscript The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain *Manuscript 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 Lattice Dynamics Calculations based on Density-functional Perturbation Theory in Real Space Honghui Shanga , Christian Carbognoa , Patrick Rinkea,b , Matthias Schefflera a Fritz-Haber-Institut b COMP/Department der Max-Planck-Gesellschaft, Faradayweg 4–6, D-14195 Berlin, Germany of Applied Physics, Aalto University, P.O Box 11100, Aalto FI-00076, Finland Abstract A real-space formalism for density-functional perturbation theory (DFPT) is derived and applied for the computation of harmonic vibrational properties in molecules and solids The practical implementation using numeric atom-centered orbitals as basis functions is demonstrated exemplarily for the all-electron Fritz Haber Institute ab initio molecular simulations (FHI-aims) package The convergence of the calculations with respect to numerical parameters is carefully investigated and a systematic comparison with finite-difference approaches is performed both for finite (molecules) and extended (periodic) systems Finally, the scaling tests and scalability tests on massively parallel computer systems demonstrate the computational efficiency Keywords: Lattice Dynamics, Density-function theory, Density-functional Perturbation Theory, Atom-centered basis functions PACS: 71.15.-m Introduction Density-functional theory (DFT) [1, 2] is to date the most widely applied method to compute the ground-state electronic structure and total energy for polyatomic systems in chemistry, physics, and material science Via the Hellmann-Feynman [3, 4] theorem the DFT ground state density also provides access to the first derivatives of the total energy, i.e., the forces acting on the nuclei and the stresses acting on the lattice degrees of freedom The forces and stress in turn can be used to determine equilibrium geometries with optimization algorithms [5], to traverse thermodynamic phase space with ab initio molecular dynamics [6], and even to search for transition states of chemical reactions or structural transitions [7] Second and higher order derivatives, however, cannot be calculated on the basis of the ground state density alone, but also require knowledge of its response to the corresponding perturbation: The 2n + theorem [8] proves that the n-th order derivative of the density/wavefunction is required to determine the 2n + 1-th derivative of the total energy For example, for the calculation of vibrational frequencies and phonon band-structures (second order derivative) the response of the electronic structure to a nuclear displacement (first order derivative) is needed These derivatives can be calculated in the framework of density-functional perturbation theory (DFPT) [9–11] viz the coupled perturbed self-consistent field (CPSCF) method [12–17] DFPT and CPSCF then provide access to many fundamental physical phenomena, such as superconductivity [18, 19], phonon-limited carrier lifetimes [20–22] in electron transport and hot electron Formally, DFPT and CPSCF are essentially equivalent, but the term DFPT is more widely used in the physics community, whereas CPSCF is better known in quantum chemistry Preprint submitted to Computer Physics Communications relaxation [23, 24], Peierls instabilities [25], the renormalization of the electronic structure due to nuclear motion [26–35], Born effective charges [36], phonon-assisted transitions in spectroscopy [37–39], infrared [40] as well as Raman spectra [41], and much more [42] In the literature, implementations of DFPT using a reciprocal-space formalism have been mainly reported for plane-wave (PW) basis sets for norm-conserving pseudopotentials [9, 10, 36], for ultrasoft pseudopotentials [43], and for the projector augmented wave method [44] These techniques were also used for all-electron, full-potential implementations with linear muffin tin orbitals [45] and linearized augmented plane-waves [46, 47] For codes using localized atomic orbitals, DFPT has been mainly implemented to treat finite, isolated systems [12–17], but only a few literature reports exist for the treatment of periodic boundary conditions with such basis sets [48–50] In all these cases, which only considered perturbations commensurate with the unit cell (Γ-point perturbations), the exact same reciprocal-space formalism has been used as in the case of plane-waves Sun and Bartlett [51] have analytically generalized the formalism to account for non-commensurate perturbations (corresponding to non-Γ periodicity in reciprocalspace), but no practical implementation has been reported In the aforementioned reciprocal-space implementations, each perturbation characterized by its reciprocal-space vector q requires an individual DFPT calculation Accordingly, this formalism can become computationally expensive quite rapidly, whenever the response to the perturbations is required to be known on a very tight q grid To overcome this computational bottleneck, various interpolation techniques have been proposed in literature: For instance, Giustino et al [52] suggested to Fourier-transform the reciprocal-space electron-phonon coupling elements to real-space The spatial localization of the perFebruary 1, 2017 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 +0.00 +0.16 +0.32 +0.48 +0.64 +0.80 turbation in real-space (see Fig 1) allows an accurate interpolation by using Wannier functions as a compact, intermediate representation In turn, this then enables a back-transformation onto a dense q grid in reciprocal-space To our knowledge, however, no real-space DFPT formalism that directly exploits the spatial localization of the perturbations under periodic boundary conditions has been reported in the literature, yet This is particularly surprising, since realspace formalisms have attracted considerable interest for standard ground-state DFT calculations [53–59] in the last decades due to their favorable scaling with respect to the number of atoms and their potential for massively parallel implementations Formally, one would expect a real-space DFPT formalism to exhibit similar beneficial features and thus to facilitate calculations of larger systems with less computational expense on modern multi-core architectures We here derive, implement, and validate a real-space formalism for DFPT The inspiration for this approach comes from the work of Giustino et al [52], who demonstrated that Wannierization [60] can be used to map reciprocal-space DFPT results to real-space, which in turn enables numerically efficient interpolation strategies [61] In contrast to these previous approaches, however, our DFPT implementation is formulated directly in real space and utilizes the exact same localized, atomcentered basis set as the underlying ground-state DFT calculations This allows us to exploit the inherent locality of the basis set to describe the spatially localized perturbations and thus to take advantage of the numerically favorable scaling of such a localized basis set In addition, all parts of the calculation consistently rely on the same real-space basis set Accordingly, all computed response properties are known in an accurate real-space representation from the start and no potentially error-prone interpolation (re-expansion) is required However, this reformulation of DFPT also gives rise to many non-trivial terms that are discussed in this paper For instance, the fact that we utilize atom-centered orbitals require accounting for various Pulay-type terms [62] Furthermore, the treatment of spatially localized perturbations that are not translationally invariant with respect to the lattice vectors requires specific adaptions of the algorithms used in ground-state DFT to compute electrostatic interactions, electronic densities, etc We also note that the proposed approach facilitates the treatment of isolated molecules, clusters, and periodic systems on the same footing Accordingly, we demonstrate the validity and reliability of our approach by using the proposed real-space DFPT formalism to compute the electronic response to a displacement of nuclei and harmonic vibrations in molecules and phonons in solids The remainder of the paper is organized as follows In Sec we succinctly summarize the fundamental theoretical framework used in DFT, in DFPT, and in the evaluation of harmonic force constants Starting from the established real-space formalism for ground-state DFT calculations, we derive the fundamental relations required to perform DFPT and lattice dynamics calculations in section The practical and computational implications of these equations are then discussed in Sec using our own implementation in the all-electron, full-potential, numerical atomic orbitals based code FHI-aims [55, 63, 64] as an - 0.20 - 0.12 - 0.04 +0.04 +0.12 +0.20 Figure 1: Periodic Electronic density n(r) and spatially localized response of the electron density dn(R)/dRI to a perturbation viz displacement of atom ∆RI shown exemplarily for an infinite line of H2 molecules example In Section we validate our method and implementation for both molecules and extended systems by comparing vibrational and phonon frequencies computed with DFPT to the ones computed via finite-differences Furthermore, we exhaustively investigate the convergence behavior with respect to the numerical parameters of the implementation (basis set, system sizes, integration grids, etc.) and we discuss the performance and scaling with system size Eventually, Sec summarizes the main ideas and findings of this work and highlights possible future research directions, for which the developed formalism seems particularly promising Fundamental Theoretical Framework 2.1 Density-functional theory In DFT, the total energy is uniquely determined by the electron density n(r) E KS = T s [n] + Eext [n] + E H [n] + E xc [n] + Eion−ion , (1) in which T s is the kinetic energy of non-interacting electrons, Eext the electron-nuclear, E H the Hartree, E xc the exchangecorrelation, and Eion−ion the ion-ion repulsion energy All energies are functionals of the electron density Here we avoid an explicitly spin-polarized notation, a formal generalization to collinear (scalar) spin-DFT is straightforward The ground state electron density n0 (r) (and the associated ground state total energy) is obtained by variationally minimizing Eq (1) δ E KS − µ δn n(r) dr − Ne =0, (2) whereby the chemical potential µ = δE KS /δn ensures that the number of electrons Ne is conserved This yields the KohnSham single particle equations hˆ KS ψi = tˆs + vˆ ext (r) + vˆ H + vˆ xc ψi = i ψi , (3) for the Kohn-Sham Hamiltonian hˆ KS In Eq (3) tˆs is the single particle kinetic operator, vˆ ext the (external) electron-nuclear potential, vˆ H the Hartree potential, and vˆ xc the exchangecorrelation potential Solving Eq (3) yields the Kohn-Sham 2 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 single particle states ψi and their eigenenergies i The single particle states determine the electron density via f ( i )|ψi (r)|2 , n(r) = However, for higher order derivatives of the total energy, e.g., the Hessian, d2 Etot dRI dR J (4) i = in which f ( i ) denotes the Fermi-Dirac distribution function To solve Eq (3) in numerical implementations, the KohnSham states are expanded in a finite basis set χµ (r) ψi (r) = Cµi χµ (r) , (5) (6) S µνCνi i ν Using the bra-ket notation < | > for the inner product in Hilbert space, Hµν denotes the elements χµ |hˆ KS |χν of the Hamiltonian matrix and S µν the elements χµ |χν of the overlap matrix Accordingly, the variation with respect to the density in Eq (2) becomes a minimization with respect to the expansion coefficients Cνi Etot   = E KS [n0 (r)] = E KS − Cνi i = = dEtot dRI ∂Etot − − ∂RI   f ( i ) i ( ψi |ψi − 1) , dhˆ KS ∆R J , hˆ KS (∆R J ) = hˆ (0) KS + dR J µ µi ∂Etot ∂Cµi ∂Cµi ∂RI (10) hˆ (1) KS of the original Hamiltonian hˆ (0) KS We then expand the wave (1) functions ψi (∆R J ) = ψ(0) +ψ (∆R J ) and eigenvalues i (∆R J ) = i i (0) (1) i + i (∆R J ) linearly and apply the normalization condition ψi (∆R J )|ψi (∆R J ) = From the perturbed Kohn-Sham equations hˆ KS (∆R J ) |ψi (∆R J ) = i (∆R J ) |ψi (∆R J ) , (11) − ∂Etot ∂χµ − ∂χµ ∂RI µi ∂FI ∂Cµi , ∂Cµi ∂R J 2.2 Density-functional perturbation theory To determine the ∂Cµi /∂R J and ∂χµ /∂R J needed for the computation of the Hessian (Eq 9), we assume that the displacement from equilibrium ∆R J only results in a minor perturbation (linear response) (7) in which the eigenstates ψi are constrained to be orthonormal Typically, the ground state density n0 (r) and the associated total energy Etot are determined numerically by solving Eq (7) iteratively, until self-consistency is achieved To determine the force FI acting on nucleus I at position RI in the electronic ground state, it is necessary to compute the respective gradient of the total energy, i.e., its total derivative [65– 67] FI µ ∂FI ∂χµ − ∂χµ ∂R J the last term no longer vanishes since the forces are not variational with respect to the expansion coefficients Cµi Accordingly, a calculation of the Hessian does not only require the analytical derivatives appearing in the first two terms, but also the response of the expansion coefficients and the basis functions to a nuclear displacement (∂Cµi /∂R J and ∂χµ /∂R J , respectively) More generally, according to the (2n + 1) theorem, knowledge of the n-th order response (i.e the n-th order total derivative) of the electronic structure with respect to a perturbation is required to determine the respective (2n + 1)-th total derivatives of the total energy [8] These response quantities are, however, not directly accessible within DFT, but require the application of first order perturbation theory using the expansion coefficients Cµi In this expansion, Eq (3) becomes a generalized algebraic eigenvalue problem HµνCνi = (9) µ ν d FI dR J ∂FI − − ∂R J − = we then immediately obtain the Sternheimer equation [68] (8) (hˆ (0) KS − (0) (1) i ) |ψi = −(hˆ (1) KS − (1) (0) i ) |ψi (12) The corresponding first order density is given by =0 In Eq (8) we have used the notation ∂/∂RI to highlight partial derivatives The first term in Eq (8) describes the direct dependence of the total energy on the nuclear degrees of freedom The second term, the so-called Pulay term [62], captures the dependence of the total energy on the basis set chosen for the expansion in Eq (5) It vanishes for a complete basis set or if the chosen basis set does not depend on the nuclear coordinates, e.g., in the case of plane-waves The last term vanishes, if Eq (7) has been variationally minimized with respect to the expansion coefficients Cµi to obtain the ground state total energy and density That this holds true also in practical numerical implementations is demonstrated in Sec Appendix A (1) ∗(1) (0) f ( i ) ψ∗(0) i (r)ψi (r) + ψi (r)ψi (r) n(r)(1) = (13) i To solve the Sternheimer equation (Eq 12), we use the DFPT formalism [9, 11] and thus the same expansion for ψ(1) i as used (0) in Eq (5) for ψi , which gives (1) (0) (0) (1) Cµi χµ (r) + Cµi χµ (r) ψ(1) i (r) = (14) µ (1) To determine the unknown coefficients Cµi , it is necessary to iteratively solve Eq (12) until self-consistency is achieved This 3 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 The linear term in this expansion is not noted because it vanishes at the equilibrium positions The Hessian in the second term (often referred to as force constants) can be determined with DFPT as described in the previous section The equaharm tions of motions for the nuclei in this potential Etot ({RI }) are analytically solvable and yield a superposition of independent harmonic oscillators for the displacements from equilibrium ∆RI (t) = RI (t) − R0I In the complex plane, these displacements correspond to the real part of      (19) Aλ exp(iωλ t) [eλ ]I  , ∆RI (t) = Re  √ MI λ Figure 2: Illustration of the atomic coordinates in the unit cell RI , its lattice vectors Rm , and the atomic coordinates in a supercell RIm = Rm + RI in which the complex amplitudes (and phases) Aλ are dictated by the initial conditions; the eigenfrequencies ωλ and the individual components [eλ ]I of the eigenvectors eλ are given by the solution of the eigenvalue problem: is best done in matrix form: ν (0) (0) (1) i S µν )C νi (0) (Hµν − =− (0) (1) (0) i S µν C νi − (1) (0) i S µν p (0) (1) Cµp U pi , (C (0)† S (1)C (0) E (0) − C (0)† H (1)C (0) ) pi (0) p − (16) whereby Rm denotes an arbitrary linear combination of a1 , a2 , and a3 (see Fig 2) Accordingly, also the size of the Hessian becomes in principle infinite, since also vibrations that break the perfect translational symmetry need to be accounted for This problem can be circumvented by transforming the harmonic force constants Φharm Im,J into reciprocal space Formally, this transforms this problem of infinite size into an infinite number of problems of finite size [69] (0) i (17) Here, the † is used to denote the respective Hermitian conjugate of the matrices, and E (0) denotes the diagonal matrices containing the eigenvalues i 2.3 The harmonic approximation: Molecular vibrations and phonons in solids DFPT is probably most commonly applied to calculate molecular vibrations or phonon dispersions in solids in the harmonic approximation, although its capabilities extend much beyond this [42] Since we will later use vibrational and phonon frequencies to validate our implementation, we will now briefly present the harmonic approximation to nuclear dynamics To approximately describe the dynamics for a set of nuclei {RI }, the total energy Eq (7) is Taylor-expanded up to second order around the nuclei’s equilibrium positions {R0I } (harmonic approximation) harm Etot ≈ Etot ({RI }) = Etot ({R0I }) + I,J (21) A technical complication arises for periodic solids, which are characterized by a translationally invariant unit cell defined by the lattice vectors a1 , a2 , and a3 Each of the N atoms RI in the primitive unit cell thus has an infinite number of periodic replicas RIm = RI + Rm , (22) whereby the respective expansion U pi coefficients are given by U pi = Φharm d2 Etot = √ DI J = √ I J MI M J MI M J dRI dR J for the dynamical matrix Cνi(0) Formally, DFPT and CPSCF are equivalent and only differ in the way the first order wave function coefficients C (1) are obtained In the DFPT formalism, C (1) is calculated directly by solving Eq (15) self-consistently In the CPSCF formalism, the coefficients C (1) are further expanded in terms of the coefficients of the unperturbed system [12, 13] (1) Cµi = (20) ν (1) Hµν − ν De = ω2 e , (15) DI J (q) = √ MI M J = √ MI M J m Φharm Im,J exp (iq · Rm ) m d2 Etot exp (iq · Rm ) , dRIm dR J (23) since the finite (3N ×3N) dynamical matrix D(q) would in principle have to be determined for an infinite number of q-points in the Brillouin zone Its diagonalization would produce a set of 3N q-dependent eigenfrequencies ωλ (q) and -vectors eλ (q) Furthermore, the displacements defined in Eq (19) acquire an additional phase factor:     i ω (q)t+q·R [ ]  λ m ∆RIm (t) = Re  √ Aλ (q)e eλ (q) I  (24) MI λ,q d2 Etot (RI − R0I )(R J − R0J ) (18) dRI dR J In reciprocal-space DFPT implementations [9, 10, 36, 47, 70], perturbations that are incommensurate with the unit 4 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 cell (q 0) are typically directly incorporated into the DFPT formalism itself For instance, a perturbation vector uλ (q) Im eλ (q) I exp (iq · Rm ) √ MI = density n(r) into individual atomic multipoles to treat the electrostatic interactions in a computationally efficient manner Accordingly, (25) VIes,tot (r − RI ) = VIes (r − RI ) − leads to a density response dn(r + Rm ) dn(r) n(1) (r + Rm ) = = exp(iqRm ) , duλ (q) duλ (q) (26) V es (r) = I dn(r + Rm ) dn(r) = , duλ (q, r) duλ (q, r) FI = − FPI = −2 fi εi − [n(r)v xc (r)]dr + E xc (n) n(r) − n MP (r) [ − − ZI [VIes,tot (0) + J I I = i (29) V Jes,tot (|R J − RI |)] ψi | tˆs |ψi + E xc [n] + − (30)    MP  es,tot n(r) − n (r)  VI (|r − RI |) dr I    es  es,tot  ZI VI (0) + V (|R J − RI |) I i,µ,ν ∂χµ (r) (hˆ ks − i )χν (r) dr , ∂RI (35) n(r) − n MP (r) ∂VIes,tot (r − RI ) dr ∂RI (36) J I → and I Jm I0 J I → (37) Jm I0 Given that the extent of our atom-centered basis set is confined [55], only a finite number of periodic images needs to be accounted for in these sums, since only a finite number of periodic images feature atomic orbitals that have non-zero overlap with the orbitals of the atoms in the unit cell, as sketched J I (33) 3.2 Periodic boundary condition To treat extended systems with periodic boundary conditions in a real-space formalism, the equations for the total-energy and the forces given in the previous section need to be slightly adapted The general idea follows this line of thought: A periodic solid is characterized by a (not-necessarily primitive) unit cell that contains atoms at the positions RI , whereby the lattice vectors a1 , a2 , a3 characterize the extent of this unit cell and impose translational invariance To compute the properties of such a unit cell, it is not sufficient to only consider the mutual interactions between the electronic density n(r) and atoms RI in the unit cell, but it is also necessary to account for the interactions of the Nuc atoms in the unit cell with the respective periodic images of the atoms RIm and of the density n(r + Rm ) = n(r), as introduced and discussed in Eq (22) Accordingly, the double sum in Eq (29) and the single sum in Eq (34) become to determine the Kohn-Sham energy E KS entering Eq (7) during the self-consistency cycles Here, v xc = δEδnxc is the exchange-correlation potential and E xc [n] is the exchangecorrelation energy For a fully converged density, the HarrisFoulkes formalism is equivalent to [55] E KS ∗ fiCµi Cνi FIMP = − VIes,tot (|r − RI |)]dr I dEtot = FHF + FPI + FIMP , I dRI and the force arising from the multipole correction is 3.1 Total energies and forces in a real-space formalism In practice, FHI-aims uses the Harris-Foulkes total energy functional [71, 72] i (32) The Pulay force can be written as DFT, DFPT, and Harmonic Lattice Dynamics in Realspace = n(r ) dr , |r − r | can be split into three individual terms The HellmannFeynman force is   es  ∂VI (0) ∂V Jes,tot (|RI − R J )|  HF   (34) + FI = ZI  ∂RI ∂RI J I (28) so that also q perturbations become tractable within the original, primitive unit cell, which is computationally advantageous However, one DFPT calculation for each q point is required in such cases In our implementation, we take a different route by choosing a real-space representation, as discussed in detail in the next section E KS VIes (r − RI ) = and nuclear contributions The respective forces (27) the translational periodicity of the unperturbed system can be restored n(1) (r + Rm ) = (31) is the full electrostatic potential stemming from atom I, which includes the electronic that is not commensurate with the primitive unit cell By adding an additional phase factor to the perturbation uλ (q, r) = uλ (q) exp (−iqr) , ZI , |r − RI | J I In both Eq (29) and here, ZI is the nuclear charge, and n MP (r) the multipole density obtained from partitioning the 5 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 System Polyethylene Graphene Si (diamond) atoms in DFPT supercell 66 242 686 C in Fig In practical calculations, these periodic images are accounted for explicitly by the construction of superclusters that encompass all N sc atoms with non-vanishing overlap with with the orbitals of the Nuc atoms in the original unit cell (see Fig 3) As discussed in detail in Ref [55, 73], also the basis set needs to be adapted to reflect the translational symmetry Since each local atomic orbital χµ (r) in Eq (5) is associated with an atom I(µ), we first introduce periodic images χµm (r) = χµ (r − RI(µ) + Rm ) for them as well Following the exact same reasoning as in Sec 2.3, the atomic orbitals used for the expansion of the eigenstates (5) are then replaced by Bloch-like generalized basis functions C A atoms in DFT supercluster 66 200 368 Table 1: Number of atoms in the unit cell and the corresponding number of atoms in the supercluster used in the ground-state DFT calculations (atoms in the red box in Fig 3) and in the DFPT supercell (black box in Fig 3) Please note that in the case of Si the increased number of atoms in the DFPT supercell originates from the fact that in this case the circle-like DFT supercluster is encompassed by an oblique DFPT supercell with the same shape as the primitive unit cell of the diamond structure DFT supercluster B atoms in unit cell 2 unit cell χµm (r) exp (−ikRm ) , ϕµ,k (r) = (38) m so that all matrix elements | become k-dependent, e.g., D Hµν (k) DFPT supercell = ϕµ,k | hˆ ks |ϕν,k e(−ik·[Rn −Rm ]) = uc m,n Figure 3: Sketch of the real space approach for the treatment of periodic boundary conditions: The blue square indicates the unit cell, which contains one blue atom (label A) The blue dashed line shows the maximum extent of its orbitals To treat periodic boundary conditions in DFT in real space, it is necessary to construct a supercluster (red solid line) which includes all periodic images that have non-vanishing overlap with the orbitals of the atoms in the original unit cell, as exemplarily shown here for atom A and B In practice, it is sufficient to carry out the integration in the unit cell alone, since translational symmetry then allows to reconstruct the full information, as discussed in more detail in Sec 3.2 and In turn, only the dark grey atoms that have non-vanishing overlap with the unit cell need to be accounted for in the integration, as exemplarily shown here for atom C The DFPT supercell highlighted in black is the smallest possible supercell that encompasses the DFT supercluster and exhibits the same translational Born-von K´arm´an periodicity as the original unit cell Accordingly, it contains slightly more atoms than the DFT supercluster, e.g., atom D (39) χµm (r) hˆ ks χνn (r)dr Please note that for practical reason the integration has been restricted to the unit cell (uc) in this case To reconstruct the full information, e.g., of the Nuc × N sc overlap matrix, the double sum and the associated phase factors run over all periodic images N sc ×N sc , whereby only atoms with non-vanishing overlap in the unit cell contribute (see Fig and Ref [73]) These sums are finite, since all basis functions are bounded by a confinement potential [55] In the expression for the Kohn-Sham energy (Eq (29)) and the Pulay force (Eq (35)), the sum over electronic states now also runs over the Nk k-points i → Nk (40) i,k Formally, the infinite periodic solid is thus treated in realspace within a finite DFT “supercluster” (see Fig 3), which explicitly includes all periodic images RIm that have nonvanishing orbital overlap with the unit cell Thereby, Eq (38) enforces the translational symmetries to be retained Accordingly, this real-space formalism for periodic solids leads to a notable, but reasonably tractable computational overhead for DFT calculations, e.g., when comparing calculations with N primitive atoms in a unit cell to calculations with N atoms 6 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 in an isolated molecule This becomes immediately evident from Tab 1, which lists some typical supercell sizes that are used in the ground state total energy calculations at the DFT level for representative 1D, 2D, and 3D systems However, the fact that the underlying DFT formalism explicitly accounts for all periodic images RIm turns out to even be advantageous in DFPT calculations For instance, the computation of the dynamical matrix in Eq (23) explicitly requires the derivatives with respect to all periodic replicas RIm As discussed in detail in the Sec 3.3, the real-space formalism allows to reconstruct all the necessary, non-vanishing elements of the Hessian that enter Eq (23) within one DFPT run In turn, this allows us to exactly compute the dynamical matrix (Eq (23)) – and thus all eigenvalues ω2λ (q) and -vectors eλ (q) – at arbitrary q-points by simple Fourier transforms In practice, we achieve this goal by computing the Hessian in a slightly larger Born-von K´arm´an [69] DFPT supercell that encompasses the supercluster used for DFT ground state calculations (cf Fig 3) By these means, the minimum image convention associated with translational symmetry can be straightforwardly exploited also in the case of perturbations that break the original symmetry of the crystal It should be noted that, for semiconductors and insulators, the size of the DFPT supercell is typically determined by the extent of the orbitals However, for metals, this may not be enough since a large number of k-points is required for convergence To be consistent with this finer k-mesh, the DFPT supercell would have to be extended to a much larger size for metals The traditional reciprocal space approach [9–11] might therefore be computationally advantageous for metal For this reason, we only apply our real-space formalism to semiconductors and insulators in the following sections of the Hellmann-Feynman force yields ΦHF I s,J d ∂V Jes (0) δI s,J0 dR J ∂R J (42)    d ∂V Jes,tot (|RI s − R J )|   − δI s,J0 , −ZI  dR J ∂RI s in which δI s,J0 = δI J δ s0 denotes a multi-index Kronecker delta To determine the total derivative of the Pulay force, we first split Eq (35) into two terms   ∂χµm (r) Pµm,νn FPIs = −2  hˆ ks χνn (r) dr (43) ∂RI s µm,νn   ∂χµm (r) − Wµm,νn (44) χνn (r) dr , ∂RI s µm,νn using the density matrix Pµm,νn = Nk ∗ f ( i )Cµi (k)Cνi (k) exp (ik · [Rm − Rn ]) , (45) i,k and the energy weighted density matrix Wµm,νn = Nk ∗ f ( i ) i (k)Cµi (k)Cνi (k) exp (ik · [Rm − Rn ]) , i,k (46) which also incorporate the phase factors arising due to periodic boundary conditions Using this notation, the total derivative of the Pulay term can be split into four terms for the sake of readability: P−H P−W P−S ΦPIs,J = ΦP−P I s,J + ΦI s,J + ΦI s,J + ΦI s,J (47) The first term 3.3 Real-Space force constants calculations ΦP−P I s,J = To derive the expressions for the force constants in realspace, we will directly use the general case of periodic boundary conditions, as introduced in the previous section Analogously to Eq (33) we can split the contributions to the Hessian (or to the force constants) defined in Eq (9) into the respective derivatives of the contributions to the force Φharm I s,J = −ZI = µm,νn dPµm,νn dR J ∂χµm (r) hˆ ks χνn (r) dr , ∂RI s (48) account for the response of the density matrix Pµm,νn The second term ΦP−H I s,J = µm,νn dF J dFI s d2 Etot P =− =− = ΦHF I s,J + ΦI s,J (41) dRI s dR J dRI s dR J (49) Pµm,νn · ∂2 χµm (r) hˆ ks χνn (r) dr ∂RI s ∂R J ∂χµm (r) dhˆ ks χνn (r) dr + ∂RI s dR J ∂χµm (r) ∂χνn (r) + hˆ ks dr , ∂RI s ∂R J Please note that we have omitted the multipole term here, since its contribution is already three orders of magnitude smaller at the level of the forces Due to the permutation symmetry (ΦI s,J = Φ J,I s ) of the force constants, the order in which the derivatives are taken is irrelevant The formulas given above for the forces FI acting on the atoms in the unit cell are equally valid for the forces FI s acting on its periodic images RI s , as long as the sums and integrals in the supercell (see Fig 3) are performed using the minimum image convention In the following, we will exploit this fact so that only total derivatives with respect to the atoms in the primitive unit cell need to be taken Consequently, the total derivative (50) (51) (52) account for the response of the Hamiltonian hˆ ks (k), while the third and fourth term ΦP−W I s,J = −2 ΦP−S I s,J = −2 dWµm,νn dR J µm,νn Wµm,νn µm,νn ∂ ∂R J ∂χµm (r) χνn (r) dr , ∂RI s (53) ∂χµm (r) χνn (r) dr ,(54) ∂RI s 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 account for the response of the energy weighted density matrix Wµm,νn and the overlap matrix S µm,νn , respectively (cf Sec 4.1) Please note that in all four contributions many terms vanish due to the fact that the localized atomic orbitals χµm (r) are associated with one specific atom/periodic image R J(µ)m , which implies, e.g., ∂χµm (r) ∂χµm (r) = δ J(µ)m,I s ∂RI s ∂RI s (55) This allows us to re-index the sums over (µm, νn) in a computationally efficient, sparse matrix formalism (cf Ref [74]) Similarly, it is important to realize that all partial derivatives that appear in the force constants can be readily computed numerically, since the χµm are numeric atomic orbitals, which are defined using a splined radial function and spherical harmonics for the angular dependence [55] electronic density DFT 1st-order overlap Details of the Implementation The practical implementation of the described formalism closely follows the flowchart shown in Fig For the sake of readability we use the notation 1st-order density matrix 1st-order density M (1) = (0) dM , dRI s (56) 1st-order total electrostatic potential to highlight that in each step of the flowchart a loop over all atoms in the unit cell RI viz all periodic replicas RI s is performed to compute all associated derivatives In the following chapters, we will use subscripts i, j for occupied KS orbitals in the DFPT supercell, and a for the corresponding unoccupied (virtual) KS orbitals, and p, q for the entire set of KS orbitals in the DFPT supercell After the ground state calculation (see Sec 2.1 and Ref [55]) is completed, the first step is to compute the response of the (1) overlap matrix S (1) We then use Uai = (Appendix B) as the initial guess for the response of the expansion coefficients and determine the response of the density matrix P(1) , which then allows to construct the respective density n(1) (r) Using that, we compute the associated response of the electrostatic potential and of the Hamiltonian hˆ (1) KS In turn, all these ingredients then allow to set up the Sternheimer equation, the solution of which allows to update the response of the expansion coefficients C (1) Using a linear mixing scheme, we iteratively restart the DFPT loop until self-consistency is reached, i.e., until the changes in C (1) become smaller than a user-given threshold In the last steps, the response of the energy weighted density matrix W (1) , the force-constants ΦIm,J , and the dynamical matrix D(q) are computed and diagonalized on user-specified paths and grids in reciprocal space DFPT 1st-order Hamiltonian 1st-order expansion coefficients 1st-order energy density matrix force constants dynamical matrix Figure 4: Flowchart of the lattice dynamics implementation using a real-space DFPT formalism 4.1 Response and Hessian of the Overlap Matrix The first step after completing the ground state DFT calculation is to compute the first order response of the overlap matrix, a quantity that is not required in plane-wave implementations, but that needs to be accounted for when using localized atomic 8 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 + + Figure 5: Integration strategy for the computation of matrix elements, here shown exemplarily for the overlap matrix elements, see Eq (58) Instead of integrating over the whole space, the integration is restricted to the unit cell and the individual contributions arising from translated basis function pairs are summed up Figure 6: Integration strategy for the computation of the response matrix elements, here shown for the first order overlap matrix S (1) in Eq (60) Please note that to be able to restrict the integration to the unit cell, the derivative has to be translated together with the orbital as shown in Eq (59) orbitals [62] Using the definition of the overlap matrix S given in Eq (58), it becomes clear that the individual elements are related by translational symmetry (0) S µm,νn (0) χµm (r)χνn (r)dr = S µ(m−n),ν0 = using: (57) Therefore, it is possible to restrict the integration to the unit cell (uc) (0) S µm,ν0 = n uc χµ(m+n) (r)χνn (r)dr , ∂χµm (r) χνn (r) dr ∂RI s ∂2 χµ(m+t) (r) χν(n+t) (r)dr uc ∂RI(s+t) ∂R Jt ∂ ∂R J = t (58) + uc and to reconstruct the whole integral by summing over all periodic replicas n, as illustrated in Fig For the response of the overlap matrix, translational symmetry (1) S µm,νn = (0) ∂S µm,νn ∂RI s = (0) ∂S µ(m−n),ν0 ∂RI(s−n) Again, only a few contributions exist for the first term (59) δK(µ)m,I s δK(µ)m,J0 enables us again to restrict the integration to the unit cell (1) S µm,ν0 = uc n + uc ∂χµ(m+n) (r) χνn (r)dr ∂RI(s+n) χµ(m+n) (r) (60) δK(µ)m,I s δK(ν)n,J0 as illustrated in Fig Please note that only very few nonvanishing contributions exist, since every orbital only depends on the position of one specific atom or replica uc uc ∂2 χµ(m+t) (r) χν(n+t) dr = ∂RI(s+t) ∂R Jt uc (62) ∂2 χµ(m+t) (r) χν(n+t) ∂RI(s+t) ∂R Jt and for the second term ∂χνn (r) dr , ∂RI(s+n) ∂χµ(m+n) (r) χνn (r)dr = δ J(µ)m,I s ∂RI(s+n) ∂χµ(m+t) (r) ∂χν(n+t) (r) dr ∂RI(s+t) ∂R Jt uc , (61) ∂χµ(m+t) (r) ∂χν(n+t) (r) dr = ∂R Jt uc ∂RI(s+t) ∂χµ(m+t) (r) ∂χν(n+t) (r) dr ∂R Jt uc ∂RI(s+t) (63) 4.2 Response of the Density Matrix The first step in the DFPT self-consistency cycle is to calculate of the response of the density matrix using the given expansion coefficients C (0) and C (1) Using the discrete Fourier transform ∂χµ(m+n) (r) χνn (r)dr ∂RI(s+n) Following the same strategy, also the second order derivatives of the overlap matrix required in Eq (54) can be computed (0) Cµm,i = k (0) Cµ,i (k) exp (−ik · Rm ) , (64) 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 we get E (0) U (1) − U (1) E (0) − C (0)† S (1)C (0) E (0) = −C (0)† (1) H C (0) +E (1) (92) Thereby, we have used the orthonormality relation: C (0)† S (0)C (0) = (93) Due to the diagonal character of E (0) and E (1) , this matrix equation contains the response of the eiqenvalues on its diagonal (1) p = C (0)† H (1)C (0) − C (0)† S (1)C (0) E (0) pp (94) Conversely, the off-diagonal elements determine the response of the expansion coefficients for p q U (1) pq = (C (0)† S (1)C (0) E (0) − C (0)† H (1)C (0) ) pq (ε p − q ) (95) The orthogonality relation (1) (1) (0) Ψ(0) =0, p |Ψ p + Ψ p |Ψ p (96) then also yields the missing diagonal elements U (1) pp = − + (0)† (1) (0) C S C pp (97) 4.7 Response of the Energy Weighted Density Matrix After achieving self-consistency in the DFPT loop, the last task is to determine the response of the energy weighted density matrix (0) (0) (0) Wµm,νn = f ( i ) iCµm,i Cνn,i , (98) + i that is required for the evaluation of Eq (53) In close analogy to the density matrix formalism discussed in Sec 4.2, the response of the energy weighted density matrix can be expressed as: Figure 8: Integration strategy for the computation of the Hamiltonian matrix (0) (1) elements Hµm,ν0 and the response elements Hµm,ν0 The first row (a) shows the ground-state Kohn-Sham Hamiltonian, which –due to its periodicity– can be integrated using the exact same strategy used for the overlap matrix S (0) (see (1) Fig 5) The remaining rows (b) highlight that the response Hµm,ν0 requires to account for derivatives of the Kohn-Sham Hamiltonian dhˆ KS /dRI s , which is not periodic To restrict the integration to the unit cell, it is thus necessary to translate also this perturbation accordingly For this exact reason, a Born-von K´arm´an supercell [69] supercell is needed in DFPT, but not in the case of a periodic Hamiltonian as in DFT (1) Wµm,νn = f ( i) (1) (0) (0) i C µm,i C νn,i (1) (0) (0) (1) + iCµm,i Cνn,i + iCµm,i Cνn,i i (99) In close analogy to our discussion of the density matrix, the energy weighted density matrix is also evaluated in practice directly in terms of U (1) , as detailed in Appendix C 4.8 Symmetry of the Force Constants As mentioned above, the individual force constant elements are related to each other by translational symmetry ΦI s,J0 = ΦI(s+m),Jm , (100) and permutation symmetry ΦI s,Jm = Φ Jm,I s (101) Due to these symmetries, only a subset Nuc ×N sc of the complete N sc × N sc force constant matrix needs to be computed for a supercell containing N sc atoms (see Fig and Tab 1) Similarly, 12 3036 cm-1 5.5 2955 cm-1 -1 |ω - ω (Nr,mult 3)| [cm ] 300 -1 1421 cm |ω - ω(tier 3)| [cm-1] 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 -1 1330 cm -1 2955 cm 4.5 1421 cm-1 1330 cm-1 3.5 -1 200 3036 cm-1 785 cm 300 cm-1 785 cm -1 300 cm-1 2.5 1.5 100 0.5 0 -0.5 minimal tier tier tier Figure 9: Convergence of the infrared-active vibrational frequencies of ethane with respect to the basis set size (see text) We use really-tight grid setting with Nr,mult =2 and Nang,max =590 The benchmark values is calculated using “tier 3” ΦI s,J0 , Nr,mult Figure 10: Convergence of the infrared-active vibrational frequencies of ethane with respect to the radial grid density, as controlled by the parameter Nr,mult (see text) We use a “tier 2” basis set and Nang,max =590 here The benchmark values are calculated using Nr,mult =3 invariance under a complete translation of the system implies the so called “acoustic sum rule” Φ J0s,J0 = − and correlation (LDA parametrization of Perdew and Zunger [76] for the correlation energy density of the homogeneous electron gas based on the data of Ceperley and Alder [77]) In all cases, the DFPT calculations were performed for the respective equilibrium geometry, i.e., the structure obtained by relaxation (maximum force < 10−4 eV/Å) using the exact same computational settings Due to the fact that the exact same formalism is used for both for finite systems and periodic materials, the presented convergence studies are also valid for both cases Fig shows the absolute change in these vibrational frequencies if the basis set size is increased Here, a minimal basis (half a basis function per electron in the spin-unpolarized case) includes the orbitals that would be occupied orbitals in a free atom following the Aufbau principle Additional sets of basis functions are added in “tier 1”, “tier 2”, calculations, see Ref [55] for more details The vibrational frequencies converge quickly with the basis set size Already at a “tier 1” level we get qualitatively correct results with a maximal absolute/relative error of 18 cm−1 /0.6 % Fully quantitatively converged calculations are achieved with the “tier 2” basis set Atom-centered grids are used for the numerical integrations in FHI-aims [55]: Radially, each atom-centered grid consists of Nr spherical integration shells, the outermost of which lies at a distance router from the nucleus The shell density can be controlled by means of the radial multiplier Nr,mult For example, Nr,mult =2 results in a total of 2Nr +1 radial integration shells On these shells, angular integration points are distributed in such a way that spherical harmonics up to a certain order are integrated exactly by the use of Lebedev grids as proposed by Delley [78] Here, we characterize the angular integration grids by the maximum number of angular integration points Nang,max used in the calculation Fig 10 and Fig 11 show our convergence tests with respect to Nr,mult and Nang,max , respectively In both cases, we find that the computed vibrational frequencies depend only weakly on the chosen integration grids: For Nr,mult , even the (102) (I s) (J0) which enables us to determine the entries on the diagonal Φ J0,J0 from the off-diagonal elements For our implementation, this is computationally favorable, since no special treatment of “onsite” terms, i.e., contributions stemming from one individual atom, is required, e.g., in Eq (42) or for the integration of “onsite” matrix elements [73] Please note that space and point group symmetries [69], which would allow to further reduce the amount of force constants that need to be computed, are not exploited in the implementation, yet Validation and Results To validate our implementation we have specifically investigated the convergence of vibrational frequencies with respect to the numerical parameters used in the calculation in Sec 5.1 Furthermore, a systematic validation of the implementation by comparing to vibrational frequencies obtained from finitedifferences is presented in Sec 5.2; these tests are extended to periodic systems in Sec 5.3 Eventually, the computational performance of the implementation is discussed in Sec 5.4 5.1 Convergence with respect to Numerical Parameters First, we analyze the convergence behavior of our DFPT implementation with respect to the numerical parameters used in the calculation, i.e., the basis set size used in the expansion (Eq 5) of the Kohn-Sham states in numerical, atom centered orbitals and the (radial and angular) grids used for the numerical integration Exemplarily, we discuss these effects using the six infrared active frequencies of ethane (C2 H6 ), which in all cases are computed using a local approximation for exchange 13 finite-difference 0.007 DFPT 3036 cm-1 |ω - ω (Naug,max 590)| [cm-1] -1 0.006 2955 cm 1421 cm-1 0.005 Intensity 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 1330 cm-1 0.004 -1 785 cm 300 cm-1 0.003 0.002 0.001 400 302 434 To validate our DFPT implementation, we have compared the obtained vibrational frequencies to finite-difference calculations, in which the Hessian was obtained via a first order finitedifference expression for the forces and dipole moments (see below) using an atomic displacement of 0.0025 Å Exemplarily, we discuss the performance of our implementation using the infrared (IR) spectrum of the C60 molecule The IR intensity ∼ I = I eλI √ MI (1) n (r) r dr , (103) for a given vibrational eigenmode eλ can be computed both with finite-differences and DFPT by inspecting the changes induced in the dipole moment µ = n(r) r dr by the displacements associated with the vibrational mode λ As Fig 12 illustrates, both the IR frequencies and intensities agree very well between the finite-difference approach and our DFPT implementation To validate our DFPT implementation in a more systematic way, we have also compared the vibrational frequencies of 32 selected molecules with finite-difference calculations, utilizing the exact same first order finite-difference formalism used for the C60 molecule All calculations were performed at the LDA level of theory using fully converged numerical parameters 2 So DFPT 532.8 591.6 1220.5 1474.9 1400 Intensity (a.u.) fd 0.607 0.435 0.211 0.349 1600 DFPT 0.604 0.417 0.214 0.347 for the equilibrium geometry determined by relaxation (maximum force < 10−4 eV/Å) A detailed list of results for these calculations is given in the Appendix D For the sake of readability, we here only discuss the difference between the vibrational frequencies obtained via DFPT and via finite-differences, which we quantify by the mean absolute error (MAE), the maximum absolute error (MaxAE), the mean absolute percentage error (MAPE) and the maximum absolute percentage error (MaxAPE) for each molecule These statistical data is succinctly summarized in Tab 2: Overall, we find an excellent agreement between our DFPT implementation and the finitedifference results (average MaxAE of 0.64 cm−1 and average MaxAPE of 0.09%) Please note that the largest occurring absolute error (10.13 cm−1 in P2 ) and the largest occurring relative error (1.46% in H2 O2 ) still correspond to relatively moderate relative and absolute errors (1.26% and 5.73 cm−1 , respectively) The occurrence of these deviations are in part caused by numerical errors, e.g., the ones arising due to the moving integration grid [55] and due to the finite multipole expansion [55] (The multipole term in force constants calculation Eq.(41) has been omitted) Such errors affect these two approaches (finite difference and DFPT) differently To a large extent, this is mitigated in these benchmark calculations by choosing highly-accurate settings Still, the finite-difference reference calculations themselves exhibit a certain uncertainty, since they can be sensitive to the atomic displacement chosen for evaluating the numeric derivatives For instance, this is the 5.2 Validation against Finite-Differences 1000 1200 Frequency (cm)-1 Figure 12: IR spectrum for the C60 molecule computed at the LDA level of theory using tight grid settings, a “tier 1” basis set, and a Gaussian broadening of 30 cm−1 The finite-difference (fd) and the DFPT result lie almost on top of each other, as the exact values listed in the table below substantiate most sparse radial integrations grids yields qualitative and almost quantitatively correct frequencies, since the maximum absolute and relative errors are 5.5 cm−1 and 1.8 %, respectively Quantitatively converged results are achieved at the Nr,mult = level with absolute and relative errors of 0.2 cm−1 and 0.08 % As Fig 11 shows, the vibrational frequencies are virtually unaffected by the angular integration grids; the maximum absolute error is always smaller than 0.01 cm−1 ∂µ eλI √ ∂RI MI 800 Frequency (cm−1 ) fd 530.8 591.3 1217.0 1475.5 Figure 11: Convergence of the infrared-active vibrational frequencies of ethane with respect to the angular integration grid, as controlled by the parameter Nang,max (see text) We use a “tier 2” basis set and Nr,mult =2 here The benchmark values are calculated using Nang,max =590 IλIR 600 590 Naug,max numerical settings Additionally, we increased the order of the multipole expansion to l = 12 and the radial integration grid to Nr,mult = for all systems except LiF, NaCl, and P2 An atomic displacement of 0.013 Å was used in the finite-difference calculations called “tier 2” basis sets and “really tight” defaults were used for the 14 0.12 DOS (a.u.) k=18 Cl2 ClF CO CS F2 H2 HCl HF Li2 LiF LiH N2 Na2 NaCl P2 SiO H2 O SH2 HCN CO2 SO2 C2 H2 H2 CO H2 O2 NH3 PH3 CH3 Cl SiH4 CH4 N2 H4 C2 H4 Si2 H6 Average 0.06 0 350 700 1050 1400 2800 3000 k=3 k=11 0.06 DOS - DOS (k=18) (a.u.) 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 -0.06 350 700 Frequency (cm-1) 1050 1400 2800 3000 Figure 13: Convergence of the phonon density of states of polyethylene with respect to the number of k-points utilized in the primitive Brillouin zone for DFPT calculations of the C2 H4 chain The top panel shows the density of states for 18 k-points and the bottom panel shows the difference with respect to this converged reference A Gaussian broadening of cm−1 and 200 q points were used in the computation of g(ω) case for the P2 molecule, which exhibits the largest absolute error in Tab For this reason, we have also compared our DFPT calculations with benchmark results (Gaussian code, aug-ccpVTZ basis set) reported in the “NIST Computational Chemistry Comparison and Benchmark Database” [79] For the 15 dimers contained both in Tab and in this database, the mean absolute percentage errors is only 0.5% To showcase the ability of our implementation to treat finite systems and periodic solids on the same footing, we compare the vibrational frequencies of various polyethylene chains H(C2 H4 )n H with different lengths (n from to 8) to the respective periodic, infinite chain of C2 H4 In the latter case, we compute the vibrational/phonon density of states (DOS) Nq q λ δ[ω − ωλ (q)] , MaxAE (cm−1 ) 0.15 0.63 1.42 0.61 0.50 2.33 1.22 2.80 0.40 0.32 0.18 1.48 0.19 0.64 10.13 0.50 1.87 0.59 1.40 1.66 0.50 1.47 0.98 5.73 0.75 0.32 0.77 0.24 0.65 1.05 2.88 0.62 0.64 MAPE (%) 0.03 0.08 0.07 0.05 0.05 0.06 0.04 0.07 0.12 0.03 0.01 0.06 0.12 0.17 1.26 0.04 0.05 0.02 0.05 0.06 0.05 0.05 0.03 0.26 0.03 0.01 0.02 0.02 0.02 0.04 0.07 0.05 0.04 MaxAPE (%) 0.03 0.08 0.07 0.05 0.05 0.06 0.04 0.07 0.12 0.03 0.01 0.06 0.12 0.17 1.26 0.04 0.12 0.05 0.04 0.07 0.10 0.04 0.05 1.46 0.02 0.03 0.03 0.03 0.05 0.15 0.31 0.45 0.09 Table 2: Mean absolute error (MAE), maximum absolute error (MaxAE), mean absolute percentage error (MAPE) and max absolute percentage error (MaxAPE) for the difference between the vibrational frequencies obtained via DFPT and via finite-differences using an atomic displacement of 0.013 Å for a set of 32 molecules All calculations are performed at the LDA level of theory with fully converged numerical settings and relaxed geometries (see text and respective footnote) 5.3 Extended Systems: Phonons g(ω) = MAE (cm−1 ) 0.15 0.63 1.42 0.61 0.50 2.33 1.22 2.80 0.40 0.32 0.18 1.48 0.19 0.64 10.13 0.50 1.14 0.29 0.96 0.97 0.41 0.82 0.47 1.27 0.47 0.18 0.35 0.19 0.35 0.54 0.70 0.18 0.43 (104) 15 whereby a normalized Gaussian function with a width σ of cm−1 is used to approximate the Delta-distribution δ[ω − ωλ (q)] It should be noted that the phonon DOS of an infinite C2 H4 chain is not zero at the Γ-point, because it is a one-dimensional system [80] All calculations have been performed for relaxed equilibrium geometries (maximum force < 10−4 eV/Å) with fully converged numerical parameters, i.e., using the aforementioned really-tight integration grids and “tier 2” basis sets For the periodic chain, a reciprocal-space grid of 11×1×1 electronic k-points and a grid of 200×1×1 vibrational q-points (in the primitive Brillouin zone) has been utilized to converge the density of states g(ω), as substantiated in Fig 13 and Fig 14 Whereas the convergence with respect to electronic k-points is reasonably fast, a large amount of vibrational q-points is required to sample the Brillouin zone, especially for the relatively moderate broadening σ of cm−1 In this context, it is important to realize that the actual number of q-points used is not at all computationally critical in our implementation: As discussed in Sec 2.3, our implementation involves determining all non-vanishing force-constants in real-space; the respective q-dependent properties can then be determined exactly by a simple Fourier transform with minimal numerical effort For instance, using q = 2000 only requires ∼ s more computational time than the q = 20 case The outcome of these investigations is summarized in Fig 15, in which the vibrational density of states (σ=1 cm−1 ) for the isolated H(C2 H4 )n H chains with variable length (n from to 8) is compared to the vibrational density of states (σ=5 cm−1 ) of the extended, infinitely long polyethylene (C2 H4 ) chain With increasing length n, the vibrational frequencies of the isolated chain start to resemble the density of states g(ω) of the infinitely long polyethylene chain Still, some features, e.g., the low frequency modes that stem from long-wavelength phonons can only be correctly captured in the periodic DFPT calculation Please note that the differences between the vibrational density of state of the H(C2 H4 )8 H molecule (50 atoms) and the C2 H4 chain (66 atoms in the DFPT supercell) are to a large extend not caused by the additional force-constants accounted for in the periodic case Rather, the differences stem from the fact that the molecular vibrational density of states effectively corresponds to a reciprocal-space sampling of q ≈ 8, which –as Fig 14 shows– is not sufficient to capture the contributions of long-range wavelengths to the density of states Eventually, we have also validated our real-space implementation against finite-difference calculations performed using phonopy [81, 82] for two realistic periodic systems As a two-dimensional example, we use graphene, the vibrational properties of which have been controversially debated in the literature [83, 84], especially regarding the role of long-ranged interactions that are not treatable in real-space As discussed in Sec 4.4 already, correction terms that can account for such interactions are not yet part of the implementation discussed in this work To avoid possible artifacts due these effects, we have thus performed finite-difference calculations (displacement 0.008 Å) in the exact same 11 × 11 × supercell (242 atoms) that is also inherently used in the DFPT calculations itself (see Fig 3) In both the case of DFPT and finite- 0.12 DOS (a.u.) q=2000 0.06 0 350 700 1050 1400 2800 3000 0.03 q=20 DOS - DOS (q=2000) (a.u.) 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 q=200 0.02 0.01 -0.01 350 700 -1 Frequency (cm ) 1050 1400 2800 3000 Figure 14: Same as Fig 13, but for the convergence with respect to the number of q-points in the primitive Brillouin zone A Gaussian broadening of cm−1 is used 16 Graphene 2500 finite-difference DFPT 2000 Frequency (cm-1) 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 1500 1000 500 Γ K M Γ Figure 16: Vibrational band structure of graphene computed at the LDA level using both DFPT (solid blue line) and finite-difference (red open circles) All calculations have been performed using a 11×11×1 k-grid sampling for the primitive Brillouin zone, tight settings for the integration, and a “tier 1” basis set 600 differences, all calculations have been performed for relaxed equilibrium geometries (maximum force < 10−4 eV/Å) with 11×11×1 k-points in the primitive unit cell, tight settings, the “tier 1” basis set, and the LDA functional As Fig (16) shows, we find an excellent agreement between our DPFT implementation and the finite-difference calculations: By using such extended supercells, even the parabolic dispersion [85] in the lowest acoustic branch and the Kohn anomalies at Γ and K are captured in a qualitatively correct fashion by both our realspace DFPT and the finite-difference approach, as shown by Maultzsch et al [83] before Our implementation is thus ideally suited to further investigate to which extent long-range corrections to the perturbation potential will alter these effects For a three-dimensional system, we have used silicon in the diamond structure as an example All calculations have been performed for relaxed equilibrium geometries (maximum force < 10−4 eV/Å) with 7×7×7 k points in the primitive Brillouin zone, tight settings for the integration, a “tier 1” basis set, and the LDA functional Finite-difference calculations have been performed again using phonopy [81, 82] with a × × supercell of the conventional cubic fcc cell (1000 atoms) and a finite displacement of 0.01 Å, which yields fully converged vibrational band structures (error < cm −1 ) This was systematically checked by running finite-difference calculations for up to × × supercells of the primitive unit cell (1458 atoms) As shown in Fig (17), our DPFT implementation again yields an excellent agreement with the respective finite-difference calculations 1200 2800 3100 Figure 15: Vibrational frequencies for increasingly longer H(C2 H4 )n H chains compared to the vibrational density of states g(ω) of an infinite C2 H4 chain All calculations were performed using the LDA functional and with converged numerical parameters (see text) Already for a length of n=8, the vibrational frequencies of the isolated chain start to resemble the density of states g(ω) of the infinitely long polyethylene chain (bottom panel) 5.4 Performance and Scaling of the Implementation To systematically investigate the performance and scaling of our implementation, we here show timings for the H(C2 H4 )n H molecules with variable length n = − 90 and the polyethylene chain C2 H4 In the latter case, we have systematically increased 17 the number of building units in the unit cell from (C2 H4 )1 to (C2 H4 )12 All calculations use a “tier 1” basis set, light settings for the integrations, and the LDA functional 11 × × kpoints were used to sample the primitive Brillouin zone in the periodic case We performed all these calculations on a single node featuring two Intel Xeon E5-2698v3 CPUs (32 cores) and Gb of RAM per core For the timings of the finite molecules shown in Fig 18, we find that the integration of the Hamiltonian response matrix H (1) determines the computational time for small system sizes i.e., for less than 200 atoms As it is the case for the update of the response density n(1) , which involves similar numerical operations, we find a scaling of O(N ) for this step (see Tab 3) This is not too surprising, since these operations, which scale with O(N) at the ground-state DFT level [55], need to be performed 3N times when assessing the Hessian at the DFPT level, i.e., once for each cartesian perturbation of each atom For the exact same reasons, the treatment of electrostatic effects, which scales as ∼ O(N 1.6 ) at the ground-state DFT level [55], scales as O(N 2.4 ) for the computation of the (1) electrostatic response potential Ves,tot For very large system sizes (N 100), the update of the response density matrix P(1) becomes dominant, since it scales as O(N 3.8 ) in this regime As discussed in Sec 4, the computation of P(1) requires matrix multiplication operations, which traditionally scale O(N ), for each of the 3N individual perturbations To assess very large systems (N 1000), it would thus be beneficial to switch to a more advanced formalism for this computational step [16, 17] To understand the timings shown in Fig 19 for the periodic linear chain, it is important to realize that such periodic calculations not directly scale with the number of atoms N, as it was the case in the finite system, in which an N × N Hessian was computed Rather, the calculations are inherently performed in a supercell (see Fig 3) that features N sc atoms in total As discussed in Sec 2.3, only an N × N sc subsection of the Hessian needs to be determined Accordingly, the scaling is thus best rationalized as function of the effective number of √ atoms Ne f f = N · N sc , as shown in Fig 19 and Tab In this representation, the scaling and the respective exponents closely follow the behaviour discussed for the finite systems already with one exception: Due to the fact that a sparse matrix formalism is used in the periodic implementation (see Sec 3.3 and Ref [74]), a more favorable scaling for the construction of the density matrix response P(1) is found As also shown in the lower panel of Fig 19 and Tab 3, the scaling does however not follow these intuitive expectations if plotted with respect to the number of atoms N present in the primitive unit cell, since Ne f f , N sc , and N are not necessarily linearly related For the case of the linear chain, the number of periodic images N sc − N with atomic orbitals that reach into the unit cell should be a constant that is independent of the chain length viz number of atoms N present in the unit cell Accordingly the ratio N sc /N decreases from a value of in the primitive C2 H4 unit cell (6 atoms) to a value of N sc /N = 3, if a (C2 H4 )4 unit cell with 24 atoms is used In √ this regime, in which Ne f f is approximately proportional to N, we find a very favourable overall scaling of O(N 1.3 ), whereby neither of Silicon finite-difference DFPT Frequency (cm-1) 600 400 200 Γ X WK Γ L Figure 17: Vibrational band structure of silicon in the diamond structure computed at the LDA level using both DFPT (solid blue line) and finitedifference (red open circles) All calculations have been performed using 7×7×7 k points in the primitive Brillouin zone, tight settings for the integration, a “tier 1” basis set, and the LDA functional H(C2H4)nH molecules (n=1-90) Time per DFPT s.c.f iteration (s) 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 100000 10000 1000 100 10 14 n(1)(r) Ves,tot(1)(r) H(1) P(1) Total (DFPT) 50 98 194 542 number of atoms Figure 18: H(C2 H4 )n H molecules: CPU time of one full DFPT cycle required to compute all perturbations/responses associated with the 3(6n + 2) (3 is for three cartesian directions, 6n+2 is the number of atoms.) degrees of freedom on 32 CPU coress (see text) Following the flowchart in Fig 4, also the timings required for the computation of the individual response properties (density n(1) , (1) (1) electrostatic potential V(1) es,tot , Hamiltonian matrix H , density matrix P ) are given Here we use light settings for the integration, a “tier 1” basis set, and the LDA functional 18 the involved steps scales worse than O(N 1.7 ) For larger system sizes (N > 24), however, the scaling deteriorates The reason for this behaviour is the rather primitive and simple strategy that we have employed in the generation of the DFPT supercells to facilitate the treatment of integrals using the minimum image convention, as discussed in Sec 3.2 Effectively, these supercells are constructed using fully intact, translated unit cells – even if a considerable part of the periodic atomic images contained in this translated unit cell not overlap with the original unit cell For the case of the linear chain, the minimal possible ratio N sc /N = is thus reached in the N = 24 case and retained for all larger systems N > 24 In this limit, Ne f f becomes proportional to N, so that we effectively recover the scaling exponents found for Ne f f and for finite molecular systems (cf Tab 3) In summary, we find an overall scaling behavior that is always clearly smaller than O(N ) for the investigated system sizes both in the molecular and the periodic case For the periodic case, we find a particularly favorable scaling regime of O(N 1.3 ) for small to medium sized unit cells N 24 As discussed in more detail in the outlook, this regime can be potentially improved and extended to larger unit cell sizes Please note that the scaling relations discussed above for the linear chain are qualitatively also found in the case of 2D and 3D materials Given that the utilized atomic orbitals are spatially confined within a cut-off radius [55], similar relations between N sc and N are effectively found in the case of graphene and silicon Although the prefactors depend on the shape and dimen√ sionality of the unit cell, the relation Ne f f ∝ N also approximately holds in these cases In this context it is very gratifying to see that even quite extended systems (molecules with more than 100 atoms and periodic solids with more than 50 atoms in the unit cell) are in principle treatable within the relatively moderate CPU and memory resources offered by a single stateof-the-art workstation Eventually, let us note that a parallelization over cores viz nodes is already part of the presented implementation, given that the discussed real-space DFPT formalism closely follows the strategies used for the parallelization of ground-state DFT calculations in FHI-aims [55, 63]: The parallelization of the operations performed on the real-space grid closely follows the strategy described in [63]; For the matrix operation, MPI based ScaLapack routines have been used to achieve a reasonable performance both regarding computational and memory parallelization The parallel scalability for a unit cell containing 1024 Si atoms is shown in Fig.20 All calculations use a “tier 1” basis set, light settings for the integrations, and the LDA functional One k-point is sufficient to sample the reciprocal space due to the large unit cell Here we give the CPU time required for one single perturbation (one atom and one cartesian coordinate) Clearly, almost ideal scaling is achieved Time per DFPT s.c.f iteration (s) Polyethylene Chain 1000 100 10 18 27 42 n(1)(r) Ves,tot(1)(r) H(1) P(1) Total (DFPT) 83 167 Neff= ( N x Nsc)1/2 Polyethylene Chain Time per DFPT s.c.f iteration (s) 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 1000 100 (1) 10 n (r) Ves,tot(1)(r) H(1) P(1) Total (DFPT) 12 24 48 96 number of atoms in unit cell Figure 19: Linear polyethylene (C2 H4 )n chain: CPU time per DFPT cycle on 32 CPU cores as a function of the effective number of atoms Ne f f (see text) in the upper panel and as function of the number of atoms present in the unit cell (lower panel) Following the flowchart in Fig 4), also the timings required for the computation of the individual response properties (density n(1) , electro(1) (1) static potential V(1) es,tot , Hamiltonian matrix H , density matrix P ) are given Here we use light settings for the integration, a “tier 1” basis set, and the LDA functional n(1) V(1) es,tot H(1) P(1) Total H(C2 H4 )n H N 2.0 2.4 2.0 3.8 2.6 Ne f f 2.0 2.4 2.2 2.7 2.4 C2 H4 chain N 24 N > 24 1.7 2.0 1.0 2.8 1.4 2.0 1.2 3.3 1.3 2.5 Table 3: Fitted CPU time exponents α for the H(C2 H4 )n H molecules (n=8-90) and the periodic polyethylene chain C2 H4 discussed in the text The fits were performed using the expression t = cN α for the CPU time as function of the number of atoms N viz the effective number of atoms Ne f f Conclusion and Outlook In this paper, we have derived and implemented a reformulation of density-functional perturbation theory in real-space 19 and validated the proposed approach by computing vibrational properties of molecules and solids In particular, we have shown that these calculations can be systematically converged with respect to the numerical parameters used in the computation Also, we have demonstrated that the computed vibrational frequencies are essentially equal to those obtained from finitedifferences – both for finite molecules and extended, periodic systems Comparison of our results with vibrational frequencies stemming from different codes and implementations is urgently needed, but would go beyond the scope of this work The key idea of the proposed approach relies on the localized nature of the response density in non-polar materials, which enables the treatment of perturbations directly in realspace On the one hand, this allows utilizing the computationally favorable real-space techniques developed over the last decades, e.g., massively parallel grid operations that scale O(N) [55, 63] On the other hand, the proposed approach allows us to determine the full, non-vanishing response in realspace in one DFPT run In turn, simple and numerically cheap Fourier transforms–without the need of invoking any Fourier interpolation–give access to the exact associated response properties in reciprocal-space We have explicitly demonstrated the viability of this approach for lattice dynamics calculations in periodic systems: In that case, we get fully q-point converged densities of states and vibrational band structures along arbitrary paths from one DFPT run in real-space Conversely, traditional reciprocal-space implementations would in principle have required a single DFPT run for each individual value of q In practice, this is often circumvented in reciprocalspace implementations, since efficient and accurate interpolation schemes for vibrational frequencies exist [86] For the exact same reasons, finite-difference strategies can yield accurate results even in very limited supercells [81, 82] However, this is no longer the case if more complex response properties such as the electron-phonon coupling [33, 52] need to be assessed In that case, reciprocal-space formalisms either need to sample the Brillouin zone by brute-force [33] or to rely on approximate interpolation strategies, e.g., using a Wannierization of the interactions in real-space [52] The approach discussed in this work allows to overcome these limitations and to consistently assess all these properties using the well-controlled wavefunction expansion already used in the ground-state DFT and thus potentially lays the foundation for future research directions in this field This is further substantiated by the scaling behavior discussed in the previous section Despite being a proof-ofconcept implementation that has not undergone extensive numerical optimization, we find the code to exhibit quite favorable scaling properties and a promising performance that can be even improved further For instance, the exploitation of space and point group symmetry would straightforwardly lead to significant savings in computational time, especially for highsymmetry periodic systems Along these lines, symmetry can also be used to optimize the construction of the supercell used in the DFPT calculations For the sake of simplicity, this procedure so far relies on translated images of the complete and intact unit cell For particularly large and/or oblique unit cells Time per DFPT s.c.f iteration (s) for one atoms one coord Extended Si system (1024 atoms in unit cell) n(1)(r) (1) Ves,tot (r) (1) H (1) P Total (DFPT) ideal scaling 100 10 32 64 128 512 1024 Number of CPU cores 1.1 Total (DFPT) ideal 1.05 Parallel Efficiency 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 0.95 0.9 0.85 0.8 0.75 0.7 32 64 128 512 1024 Number of CPU cores Figure 20: Parallel scalability for a unit cell containing 1024 Si atoms Here the CPU time per DFPT cycle for the perturbation of one atom in one cartesian coordinate is plotted as a function of the number of CPU cores The timings required for the computation of the individual response properties (density n(1) , (1) (1) electrostatic potential V(1) es,tot , Hamiltonian matrix H , density matrix P ) are also given Then red line corresponds to ideal scaling The parallel efficiency is shown in the lower pannel Here we use light settings for the integration, a “tier 1” basis set, and the LDA functional 20 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 this can result in a significant computational overhead, since the supercell can contain periodic images of atoms that not interact with the unit cell at all Accordingly, optimizing the supercell construction procedure can immediately lead to computational savings without loss of accuracy Following these strategies, linear scaling should be achievable [87] for large system sizes (hundred and more atoms per unit cell) This would facilitate DFPT calculations of vibrational properties and of the electron-phonon coupling for fully converged q-grids in complex systems, such as organic molecules adsorbed on surfaces For such kind of 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Forces with Respect to the Degree of Self-Consistency 1e−06 1e−07 To investigate to which extent the last term of Eq (8) really vanishes in practice, we have chosen Si (diamond structure) and Al (fcc) as examples In both cases, one atom was displaced by 0.1 Å, which results in forces on this atom in the order of 100 eV/Å and 10−1 eV/Å, respectively To investigate what happens in calculations, in which full self-consistency has not yet been reached, we have then run a series of calculations with different break conditions for the self-consistency cycle We only used the maximally allowed change in charge density as break condition and varied its value between 10−2 and 10−8 electrons For the last setting, full self-consistency is achieved: Indeed, the observed change in energy/eigenvalues in the last iteration of such fully converged calculations is 10−11 eV / 10−7 eV for Si, and 10−12 eV / 10−7 eV for Al In Fig A.21, we then show the respective convergence behaviour conv of the total energy ∆E = |Etot −Etot | and of the force on the disconv placed atom ∆F I = |FI − FI | with respect to these fully converged converged values As soon as Etot is converged, Eq (8) reveals that ∂Etot ∂Cµi (A.1) ∆FI = − ∂Cµi ∂RI µi 1e−08 1e−09 1e−10 1e−11 1e−12 1e−09 1e−08 1e−07 1e−06 1e−05 0.0001 0.001 0.01 0.1 change of charge density (Å−3) 10 ∆ FI = |FI − FIconv| (eV/Å) 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 Al Si 0.1 0.01 0.001 0.0001 1e−05 1e−06 1e−07 1e−09 1e−08 1e−07 1e−06 1e−05 0.0001 0.001 0.01 0.1 change of charge density (Å−3) is indeed the error we want to assess From Fig.A.21 we can see that as the change of charge density approaches zero, the error in the forces starts to vanish ∆F I = |FI − Fconv | = 10−6 For the I typical self-consistency settings used in FHI-aims (change in charge density < 106 ), the error in the force due to the nonfully-achieved self-consistency is thus typically smaller than meV/Å In this context, it is however important to note that in relaxation or MD calculations FHI-aims requires to specify a self-consistency break condition also for the maximum change in the forces, so that in practice these errors are well-controlled conv | Figure A.21: The convergence behaviour of the total energy ∆E = |Etot −Etot (top panel) and the forces ∆FI = |FI − Fconv | (bottom panel) I Appendix B First Order Density Matrix The sum over states in the first order density matrix can be divided into sums over occupied-occupied states, occupied23 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 unoccupied states, and unoccupied-occupied states: occ Table D.4: 16 dimers (1) (0) (0) (1) f ( i )(Cµm,i Cνn,i + Cµm,i Cνn,i ) P(1) µm,νn = Cl2 ClF CO CS F2 H2 HCl HF Li2 LiF LiH N2 Na2 NaCl P2 SiO MAE MAPE i (B.1) f( = q i (1) (0) (0) i )C µm,q U qi C νn,i f( + p i (0) (1) (0) f ( i )Cµm, j U ji C νn,i + = i j + i f( f( (0) (1) (0) j )C µm,i U i j C νn, j f( (0) (0) (1) i )C µm,i C νn, j U ji i i + + a i i f( (1) (0) (0) f ( i )Cµm,a Uai Cνn,i (0) (1) (0) f ( i )Cµm,i Cνn,a Uai (1) (0) (0) f ( i )Cµm,a Uai Cνn,i (0) (1) (0) f ( i )Cµm,i Cνn,a Uai (0) (1) (0) i )(C µm,i C νn, j (U ji f( = a a i j + a i j + i + j = a i (0) (0) (1) i )C µm,i C νn, j U ji (0) (1) (0) i )C µm,i C νn,p U pi + Ui(1) j )) j (1) (0) (0) i )C µm,a U C νn,i + i a (0) (1) (0) f ( i )Cµm,i Uai Cνn,a U (1)† + C (0)† S (1)C (0) + U (1) = (0) (0) (0)† (1) (0) f ( i )Cµm,i Cνn, S C )i j j (−C j (B.3) + i a (1) (0) (0) f ( i )Cµm,a Uai Cνn,i + i a (0) (1) (0) f ( i )Cµm,i Uai Cνn,a This means that we only need to calculate the occupied(1) unoccupied sum for Uai in the CPSCF equation, Eq (95), while the occupied-occupied part is computed from the first order overlap matrix Appendix C First order energy weighted density matrix Similar like the first order density matrix, by using Eq (93) and Eq (94), we can rewrite Eq (99) into an occupied-occupied part, occupied-unoccupied part, and an unoccupied-occupied part: occ occ (1) Wµm,νn = i (0) (0) (0)† (1) (0) f ( i )Cµm,i Cνn, H C )i j j (C (C.1) j −( i + j )(C (0)† S (1)C (0) )i j occ unocc f ( i ) i (Cµm,a UaiCνn,i + Cµm,i UaiCνn,a ) + i ab-err 15 63 1.42 61 50 2.33 1.22 2.80 40 32 18 1.48 19 64 10.13 50 1.5 rel-err(%) 0.03 0.08 0.07 0.05 0.05 0.06 0.04 0.07 0.12 0.03 0.01 0.06 0.12 0.17 1.26 0.04 0.14% Tab D.4, Tab D.5, Tab D.6, Tab D.7, Tab D.8, and Tab D.9 list the vibrational frequencies obtained via DFPT and via finite-differences for systems containing two, three, four, five, six, and eight atoms, respectively In these comparisons, the atomic displacement is set to 0.013 Å in finite-difference calculation We used a “tier 2” basis sets and Nr,mult = (except LiF, NaCl and P2 , in which Nr,mult = is used) for integration grids and l=12 for multipole expansion All calculations were performed at the LDA level for the equilibrium geometry determined by relaxation (maximum force < 10−4 eV/Å) The statistical data is succinctly summarized in Tab in the main text (B.2) So Eq (66) can be simplified as : i DFPT 562.70 805.05 2176.35 1285.37 1062.29 4174.46 2880.76 3975.59 345.46 929.81 1385.13 2395.33 163.89 374.63 794.79 1232.15 Appendix D 32 molecules’ frequencies From Eq (93), we have : P(1) µm,νn = finite-difference 562.85 805.68 2177.77 1285.98 1062.79 4176.79 2881.98 3978.39 345.86 930.13 1385.31 2396.81 164.08 375.27 804.92 1232.65 a 24 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 Table D.5: trimers H2 O SH2 HCN CO2 SO2 finite-difference 1544.38 3712.72 3821.62 1138.53 2623.11 2638.81 718.60 2152.87 3338.28 651.70 1353.10 2415.46 500.56 1156.15 1359.09 DFPT 1546.25 3711.88 3820.92 1139.12 2623.01 2638.62 718.20 2151.78 3336.88 652.05 1352.19 2413.80 501.06 1155.86 1358.64 MAE MAPE ab-err 1.87 84 70 59 10 19 40 1.09 1.40 35 91 1.66 50 29 45 0.76 rel-err(%) 0.12 0.02 0.02 0.05 0.00 0.01 0.06 0.05 0.04 0.05 0.07 0.07 0.10 0.03 0.03 Table D.7: 5-atom molecules CH3 Cl 0.05% Table D.6: 4-atom molecules C H2 H2 CO H2 O2 NH3 PH3 MAE MAPE finite-difference 631.92 719.65 719.65 2023.26 3315.34 3416.76 1140.29 1211.80 1458.40 1804.60 2764.60 2814.41 392.08 959.42 1282.80 1388.60 3640.98 3642.69 946.72 1576.61 1577.08 3392.19 3525.15 3525.66 946.27 1072.55 1072.65 2323.49 2338.77 2338.89 DFPT 631.52 719.22 719.22 2022.38 3314.01 3415.29 1140.16 1212.44 1458.69 1803.62 2764.21 2814.05 397.81 958.97 1282.90 1388.65 3640.44 3641.95 947.29 1577.17 1577.18 3391.44 3524.98 3525.00 946.59 1072.73 1072.74 2323.41 2338.63 2338.64 ab-error 40 43 43 88 1.33 1.47 13 64 29 98 39 36 5.73 45 10 05 54 74 57 56 10 75 17 66 32 18 09 08 14 25 0.64 rel-error 0.06 0.06 0.06 0.04 0.04 0.04 0.01 0.05 0.02 0.05 0.01 0.01 1.46 0.05 0.01 0.00 0.01 0.02 0.06 0.04 0.01 0.02 0.00 0.02 0.03 0.02 0.01 0.00 0.01 0.01 SiH4 CH4 MAE MAPE 0.07% 25 finite-difference 750.93 981.30 981.31 1303.79 1398.88 1399.34 2979.12 3079.15 3079.60 843.26 843.26 843.26 926.70 926.70 2165.07 2183.92 2183.92 2183.92 1247.98 1247.98 1247.98 1476.39 1476.39 2956.13 3083.56 3083.56 3083.56 DFPT 750.79 981.66 981.66 1303.67 1399.40 1399.42 2978.61 3078.82 3078.83 843.50 843.50 843.50 926.95 926.95 2165.04 2183.76 2183.76 2183.76 1248.63 1248.63 1248.63 1476.84 1476.84 2956.02 3083.51 3083.51 3083.51 ab-error 14 36 35 12 52 08 51 33 77 24 24 24 25 25 03 16 16 16 65 65 65 45 45 11 05 05 05 0.28 rel-error 0.02 0.04 0.04 0.01 0.04 0.01 0.02 0.01 0.03 0.03 0.03 0.03 0.03 0.03 0.00 0.01 0.01 0.01 0.05 0.05 0.05 0.03 0.03 0.00 0.00 0.00 0.00 0.02% 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 Table D.8: 6-atom molecules N H4 C H4 finite-difference 489.03 706.75 869.85 1140.48 1244.10 1272.66 1600.03 1608.97 3368.42 3371.10 3473.44 3478.32 795.78 929.08 940.78 1031.98 1183.38 1322.46 1394.15 1651.73 3040.77 3053.43 3117.11 3144.13 DFPT 489.48 707.80 870.67 1140.22 1244.56 1273.04 1600.02 1609.18 3367.52 3370.59 3472.72 3477.61 796.25 926.20 939.53 1029.91 1183.57 1322.65 1394.28 1651.42 3040.49 3053.38 3116.90 3143.72 MAE MAPE ab-error 45 1.05 82 26 46 38 01 21 90 51 72 71 47 2.88 1.25 2.07 19 19 13 31 28 05 21 41 0.59 rel-error 0.09 0.15 0.09 0.02 0.04 0.03 0.00 0.01 0.03 0.02 0.02 0.02 0.06 0.31 0.13 0.20 0.02 0.01 0.01 0.02 0.01 0.00 0.01 0.01 0.05% Table D.9: Si2 H6 Si2 H6 MAE MAPE finite-difference 136.34 336.78 336.81 429.87 592.25 592.41 778.41 845.98 883.04 883.22 896.42 896.54 2145.74 2149.57 2159.48 2159.64 2168.94 2169.10 DFPT 136.96 336.98 336.99 429.51 592.56 592.57 778.17 846.06 883.20 883.21 896.56 896.56 2145.56 2149.48 2159.42 2159.43 2168.92 2168.93 ab-error 62 20 18 36 31 16 24 08 16 01 14 02 18 09 06 21 02 17 0.18 rel-error 0.45 0.06 0.05 0.08 0.05 0.03 0.03 0.01 0.02 0.00 0.02 0.00 0.01 0.00 0.00 0.01 0.00 0.01 0.05% 26